Haha. See what I did there? Anyway, introductory paragraph. Obduction introduces, to me, what seems to be an interesting number system because, not only is it in base 4 but it also is not written linearly. Granted, just because it's fun to me and my mathy brain doesn't mean it is for everyone. But I'm going to try to help simplify how this whole thing works.

First, to even start understanding how this number system works, we need to first establish how numbers in other bases work. If you're a red-blooded American like me, or any human with a pulse and access to a computer for that matter, you use a base 10 number system. That means for any place in a number, we start at 0 and count up to 9. Once we add one more to that place, we add one to the next place and start back at 0.

Binary is a common example of a non-base 10 system. 0 and 1 are the only numbers used to count each place. So the first few numbers in binary are 0, 1, 10, 11, 100, 101, 110, 111, 1000, and so on. These equate to 0, 1, 2, 3, 4, 5, 6, 7, 8, and so on.

So how does the actual conversion work? Well, each digit is written as a multiple of a power of the base, starting with 0 for the 1's place. In base 10, that would mean the number 321 would be written as (3*10^2)+(2*10^1)+(1*10^0) = 300 + 20 + 1 = 321. (Remember, raising another number other than 0 to the 0th power yields a result of 1. Raising 0 to the 0 power implodes the universe.)

Similarly, in binary, the number 1010001 translates to (1*2^6)+(0*2^5)+(1*2^4)+(0*2^3)+(0*2^2)+(0*2^1)+(1*2^0) = 64 + 0 + 16 + 0 + 0 + 0 + 1 = 81.


Now, say you have a base 10 number you wish to convert into binary. This is done by dividing powers of the new base into the base 10 number, writing the result as a digit, and then dividing the next one down into the remainder. Repeat until the number is exhausted.

So let's look at 81 again. Yes, we already know it's 1010001 but from here we can see the actual conversion. The highest power of 2 that goes into 81 is 2^6 and it goes in once leaving a remainder of 17. If it goes in more than once or, in other words, equaling or exceeding our base, we need to choose a higher power. So we know we have a 7-digit binary number 1- - - - - -. We can see that 2^5, which is 32, cannot go into the remainder of 17 so we fill its place with 0, leaving 10- - - - -. 2^4 goes in once, leaving a remainder of 1, the next powers 2^3, 2^2, and 2^1 cannot divide into 1, and 2^0 goes in once. So we are left with 1010001.

For those unaware, the Babylonians used a base 60 system. That's why our clocks have 60 minutes to an hour and 60 seconds to a minute. So 11:45:36 on a clock would be a 3-digit Babylonian number measuring the number of seconds that had passed in the day.

Now that you (hopefully) understand numbers in other bases, we can talk about the Villein's number system. The most interesting thing about them, to me, is that they don't write numbers in a linear fashion. Now, while this isn't the most convenient thing in the world for a human to understand, imagine growing up your whole life knowing this number system. It might be just as intuitive for the Villein as it is for us to read boring old base 10 numbers. For those of you wanting to play along, find this guy in the game:




Anyway, just like with our numbers, Villein number have places as well. Look at the following image:


As you can see, the 1's place exists right in the center, the 4's place is above that, the 16's place in down and right, the 64's place is down and left, and the 256's place is up and left from there. In essence, they all work in a spiral pattern. Now, how are these actually digit places? Well, like with base 10 having the numbers 0-9 that make up any number, the Villein have their own building blocks for numbers:


From here, we can begin to discuss how the numbers are structured. As we can see, the representation for 1 draws a single line from the digit place to the dot up and left from it. For 2, a line from the place to the the top right and bottom right dots. And finally, for 3, a line from the place to the top left, top right, and bottom left dots.

Now the cool thing is, we can shift these around to all of the position dots to form actual numbers. So let's bring it all together in one handy little chart:


With this, you can write any number in base 4. Simply combine all the shapes to make the number you wish. For example, Say we want 15. The biggest power of 4 that divides into 15 is 4 (or 4^1) and it goes in 3 times, leaving a remainder of 3. Go down a power to 1 (or 4^0) and find it goes in 3 times. Therefore 15 is written as 3*(4^1) + 3*(4^0). Refer to chart to see this:


With what is presented to us in the game, we can continue to build on this concept until we reach 1023, for 1024 is 4^5, the next digit to which we do not have access.

Now, one other interesting thing I noticed was the Villein bridges. They are not as random as they appear. While you may have stumbled through Maray on guessing your way across the bridges, the actual solution is simplistic and beautiful. Check this screen out:



As you can see, I've built a bridge one block long with a fully constructed block and have done so by drawing the number 768. Each block on this bridge corresponds with one digit of the Villein number system and the completion of each block corresponds with whether there is a 0, 1, 2, or 3 in that place. Anything with a 2 or higher can be walked on.

So we can just consider this counting in base bridge since each state of the bridge represents a specific number. Because math is fun and plays fast and loose with the rules, y'all.

I should also point out that the doors that have a tendency to block your way function in a similar manner. They only allow for one digit and have 4 stages of completion, much like a bridge section. That's why you had to clear out the number, essentially set it to 0, in order to advance through a door.

For fun, we could look at the pattern presented to us and expand on our number matrix. Let's assume that the next circle of places starts top and center and works its way around clockwise. Then we should get something like this:


Why? Because I'm a nerd who likes math. This is not a good number system for big numbers since it takes up way too much space, though. Don't do your homework in it but impress your college math professors with it somehow.

I hope you enjoyed this. Or at the very least, I hope you learned. I have a passion for mathematics and this number system got me excited after I finally figured it out. It's very elogant yet rigid and systematic. So thank you for allowing me to share my passion for numbers with you and I hope you found this guide useful for making your way through Maray. Have a good day!